**4. Results and Discussion**

This section aims to predict and analyze graphically the behavior of a CMC-based Casson nanofluid under the impact of meaningfully related parameters with regard to the velocity, temperature, skin friction coefficient, and local Nusselt number. The ranges of parameters that are taken into consideration are the mixed parameter (λ > 0 & λ < 0), Casson parameter (β > 0), magnetic parameter (*M* > 0) and nanoparticles volume fraction (0.1 ≤ χ ≤ 0.2).

Table 1 shows the thermo-physical properties of CMC-water and the nanoparticles. The numerical results obtained were in a close agreemen<sup>t</sup> with the literature and can be seen in comparative Tables 2 and 3.


**Table 1.** Thermo-physical properties of CMC-water (0.0–0.4%) and metals nanoparticles [51].


**Table 2.** Comparison of Re1/2*Cf* with published findings by Nazar et al. [43] for several values of λ (β → <sup>∞</sup>, *M* = 0, χ = 0, Pr = 0.7).

**Table 3.** Heat transfer coefficient *Qw*(ξ) = <sup>−</sup>(∂θ/∂η)<sup>η</sup>= 0 with published findings by Nazar et al. [43] for several values of λ (β → <sup>∞</sup>, *M* = 0, χ = 0, Pr = 0.7).


Figures 2 and 3 display the influence of the mixed parameter in opposing and assisting flow cases (λ > 0 & λ < 0) on the skin friction coefficient and Nusselt number, respectively. From these figures, we found that the Al–CMC-water has the highest skin friction coefficient values in the case of assisting flow and the lowest in the case of the opposing flow. For the Nusselt number, Al–CMC-water has the highest value in both cases (λ > 0 & λ < 0) and this is due to the thermo-physical properties that the aluminum possesses. It can also be observed that, in both the cases of opposing and assisting flow, when λ increases, Re1/2*Cf* and Re−1/2*Nu* increase due to increase in the buoyancy force.

**Figure 3.** Mixed parameter versus the local Nusselt number.

In Figures 4 and 5 it can be seen that the increment in the value of nanoparticles volume fraction χ resulted in a noteworthy improvement in both the skin friction coefficient and Nusselt number. The improvement in the Nusselt number is caused by the enhancement of the density and thermal conductivity of CMC-water.

**Figure 4.** Nanoparticles volume fraction versus the local skin friction coefficient.

**Figure 5.** Nanoparticles volume fraction versus the local Nusselt number.

Figures 6 and 7 show the relationship between β and both the skin friction coefficient, and Nusselt number respectively. It's noticed that the Casson parameter β is inversely proportional to the skin friction coefficient, but it is directly proportional to the Nusselt number. Physically, when the values of β rise, the yield stress decreases and therefore the skin friction coefficient decreases.

**Figure 6.** Casson parameter versus the local skin friction coefficient.

**Figure 7.** Casson parameter versus the local Nusselt number.

Figures 8 and 9 illustrate the graphical findings of Re1/2*Cf* and Re−1/2*Nu* respectively, with various values of the magnetic parameter (*M*). It is clear that as the values of *M* grow, both the skin friction coefficient and Nusselt number decline. In fact, this decline is a result of the restraining that occurred in the fluid flow, caused by the increase in intensity of the magnetic current which curbs convection and thereby reduces the skin fraction coefficient and Nusselt number. Furthermore, these figures demonstrate that, whatever the values of parameters λ, χ, β or *M*, Al–CMC-water has the highest Re1/2*Cf* and Re−1/2*Nu*.

**Figure 8.** Magnetic parameter versus the local skin friction coefficient.

**Figure 9.** Magnetic parameter versus the local Nusselt number.

Figures 10 and 11 demonstrate the impact of the mixed parameter λ on the velocity and temperature in both cases opposing and assisting flow (λ > 0 & λ < 0). Both the cases of flow indicate that an increment in λ is accompanied by an improvement in the velocity or a decay in the temperature profiles. In fact, the growth in the mixed parameter enhances the thermal buoyancy force—and, hence the velocity increases.

**Figure 10.** Mixed parameter versus velocity.

**Figure 11.** Mixed parameter versus temperature.

Figures 12 and 13 confirmed that the effect of the nanoparticles volume fraction (χ), on both velocity and temperature, is a positive effect. A rise in χ leads to a quicker transfer of heat from the outside of the sphere to the fluid and thus aids in the augmentation of the thickness of the thermal layer due to the increase in the temperature of the fluid. In addition, the increase in χ enhances energy transmission, which increases the fluid velocity. According to Figures 14 and 15, higher values of the Casson parameter (β) cause a curb in the velocity and temperature, which is verifiable because the augmentation in β creates a resistance force that restricts the flow of the fluid, which restrains the nanofluid velocity.

**Figure 12.** Nanoparticles volume fraction versus velocity.

**Figure 13.** Nanoparticles volume fraction versus temperature.

**Figure 14.** Casson parameter versus velocity.

**Figure 15.** Casson parameter versus temperature.

Figures 16 and 17 depict the graphical findings of temperature and velocity versus the magnetic parameter (*M*), respectively. It is evident in these figures that as the value of *M* grows, the temperature increases but the velocity decreases. This phenomenon occurs when a magnetic current passes through a flowing nanofluid, which produces a kind of force known as the Lorentz force and, consequently, resists the nanofluid movement. It is worth noting that, whatever the values of parameters λ, χ, β or *M*, Silver–CMC-water is superior in terms of velocity, and we found that the Copper–CMC-water temperature was the highest.

**Figure 16.** Magnetic parameter versus velocity.

**Figure 17.** Magnetic parameter versus temperature.
